Partitioning a permutation into a minimum number of
monotone subsequences is NP-hard. We extend this
complexity result to minimum partitions into
unimodal subsequences. In graph theoretical terms
these problems are cocoloring and what we call
split-coloring of permutation graphs. Based on a
network flow interpretation of both problems we
introduce mixed integer programs; this is the first
approach to obtain optimal partitions for these
problems in general. We derive an LP rounding
algorithm which is a 2-approximation for both
coloring problems. It performs much better in
practice. In an online situation the permutation
becomes known to an algorithm sequentially, and we
give a logarithmic lower bound on the competitive
ratio and analyze two online algorithms.